SOLUTION: {{{ 9^x+6^x=2^(2x+1) }}}

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Question 560634: +9%5Ex%2B6%5Ex=2%5E%282x%2B1%29+
Found 2 solutions by ankor@dixie-net.com, richard1234:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
+9%5Ex%2B6%5Ex=2%5E%282x%2B1%29+
Just looking at this, you can see there is only one value for x that will make this work: x=0
+9%5E0%2B6%5E0=2%5E%282%280%29%2B1%29+
1 + 1 = 2^1

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
x = 0 works, but we need to prove it is the only solution. Note that we can write the equation as



Dividing both sides by 4^x,



As you can see, the LHS is a strictly increasing function of x (bijective or one-to-one), so there can only be one value of x in which the LHS equals 2. We have already found it to be x=0.